for large \(t\) , since \(\frac{1}{t}\log \frac{1}{\widetilde{q}^{\mathsf {KT}}_t(x)}= h(\frac{x}{t})+\frac{1}{2t}\log t + o(1)\)  by Stirling's approximation, provided that \(S_t\)  and \(t-S_t\)  are sufficiently large.By Stirling's approximation, \(\log B(x,y)\approx (x-\frac{1}{2})\log x + (y-\frac{1}{2})\log y - (x+y-\frac{1}{2})\log (x+y) + \frac{1}{2}\log (2\pi )\)  for \(x\)  and \(y\)  sufficiently large.
Compared to (REF ), it suffers an additional term \(O(\sqrt{(\log t)/t})\)  in the width of the confidence sequence.
We remark in passing that the law of iterated logarithm (LIL) [1]} implies that the optimal additional term is in the order of \(O(\sqrt{(\log \log t)/t})\) , as also commented in [2]}.
[2]} proposed a betting strategy based on a “predictable mixture” with the so-called stitching technique [1]} and analyzed that it achieves the optimal order \(O(\sqrt{(\log \log t)/t})\) .
blackApplying a similar idea of [5]}, a more recent version of [6]} (see footnote REF ) constructed another mixture portfolio based on a prior inspired by [7]}, and showed that it achieves the LIL based on a regret analysis.

